Polynomial Eulerian Characteristic of Nilmanifolds

The article studies bundle towers \(M^{n+1}\to M^{n}\to \dots \to S^1\), \(\geqslant 1\), with fiber \(S^1\), where \(M^n = L^n\!/\Gamma^n\) are compact smooth nilmanifolds and \(L^n\thickapprox \mathbb{R}^n\) is a group of polynomial transformations of the line \(\mathbb{R}^1\). The focus is on the well-known problem of calculating cohomology rings with rational coefficients of manifolds \(M^n\). Using the canonical bigradation in the de Rham complex of manifolds \(M^n\), we introduce the concept of polynomial Eulerian characteristic and calculate it for these manifolds.